The equipment comprises means (17, 18) for causing a very small quantity of liquid for cleaning the said cylinders to be sprayed in a controlled and uniformly distributed way on to a sufficiently long portion of cloth, which, in the area directly acted on by the said sprayed liquids and in other areas, is kept in uniform and distributed contact with a transverse portion of the surface of the cylinder to be cleaned, and which is pushed, upstream and downstream of this portion, into close and uniformly distributed contact with the cylinder by a rectilinear deformable strip (6) and by a rectilinear deformable presser (9) respectively, each of these having an originally convex surface facing the cylinder, the whole being arranged in such a way that the cleaning liquid acts through the cloth on the cylinder to be cleaned in an extended, uniform and progressive way and in a very small quantity, and in such a way that the liquid is conveniently retained in the said cleaning area by the said cloth and the said rectilinear pressure elements, so that it does not fall off and so that it is transferred to the dirt on the cylinder in a gradual and controlled way, while the presser (9) facilitates the penetration of the liquid into the direct and promotes the elimination and removal of the said dirt. The active surface of the presser (9) has a suitable lowrelief configuration for better distribution of the cleaning liquid, to provide a better mechanical action for cleaning the cylinder and to retain in its channels (209) of originally outwardly diverging shape a large quantity of the paper particles which constitute the dirt removed from the cylinder.